Editing Goldbach's weak conjecture (section)

{{short description|Solved conjecture about prime numbers}}
{{Infobox mathematical statement
| name = Goldbach's weak conjecture
| image = File:Letter Goldbach-Euler.jpg
| caption = Letter from Goldbach to Euler dated on 7&nbsp;June 1742 (Latin-German)<ref>''Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle'' (Band 1), St.-Pétersbourg 1843, [https://books.google.com/books?id=OGMSAAAAIAAJ&pg=PA125 pp. 125–129].</ref>
| field = [[Number theory]]
| conjectured by = [[Christian Goldbach]]
| conjecture date = 1742
| open problem = 
| first proof by = [[Harald Helfgott]]
| first proof date = 2013
| implied by = [[Goldbach's conjecture]]
| consequences = 
}}
In [[number theory]], '''Goldbach's weak conjecture''', also known as the '''odd Goldbach conjecture''', the '''ternary Goldbach problem''', or the '''3-primes problem''', states that

: Every [[odd number]] greater than 5 can be expressed as the sum of three [[prime number|primes]]. (A prime may be used more than once in the same sum.)

This [[conjecture]] is called "weak" because if [[Goldbach's conjecture|Goldbach's ''strong'' conjecture]] (concerning sums of two primes) is proven, then this would also be true. For if every even number greater than 4 is the sum of two odd primes, adding 3 to each even number greater than 4 will produce the odd numbers greater than 7 (and 7 itself is equal to 2+2+3).

In 2013, [[Harald Helfgott]] released a proof of Goldbach's weak conjecture.<ref name=":0" />  The proof was accepted for publication in the ''[[Annals of Mathematics Studies]]'' series<ref name="Annals of Mathematics Studies Princeton University Press">{{cite web | title=Annals of Mathematics Studies | website=Princeton University Press | date=1996-12-14 | url=https://press.princeton.edu/series/annals-of-mathematics-studies | access-date=2023-02-05}}</ref> in 2015, and has been undergoing further review and revision since; fully refereed chapters in close to final form are being made public in the process.<ref>{{Cite web|title=Harald Andrés Helfgott|url=https://webusers.imj-prg.fr/~harald.helfgott/anglais/book.html|access-date=2021-04-06|website=webusers.imj-prg.fr}}</ref>

Some state the conjecture as
:Every odd number greater than 7 can be expressed as the sum of three odd primes.<ref name=MathWorldConj>{{MathWorld|title=Goldbach Conjecture|urlname=GoldbachConjecture}}</ref>
This version excludes 7 = 2+2+3 because this requires the even prime 2. On odd numbers larger than 7 it is slightly stronger as it also excludes sums like 17 = 2+2+13, which are allowed in the other formulation. Helfgott's proof covers both versions of the conjecture. Like the other formulation, this one also immediately follows from Goldbach's strong conjecture.